US-20 is a cake walk. It starts off as a freeway, changes to divided highway for a while, and then into a 2/4 lane highway up easy grades with plenty of passing lanes and space. No sharp curves. I've only driven it in the Civic, but I would take a trailer or large motorhome without hesitation.
We stayed in West Yellowstone. There are things to see in West Yellowstone, but bring your own food. I found the restaurant situation to be the worst. Everything was outrageously priced - even for a remote tourist town it was expensive.
The drive from wherever we were in the park back to West Yellowstone grew tedious after a few days. If I were to go again, I would choose to stay in the park.
As littlemo pointed out, there is lots to see. You won't want to miss the Tetons either, which is just south of Yellowstone. Most people around here make some kind of loop to take in both.